| Curve name |
$X_{121p}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{121}$ |
| Curves that $X_{121p}$ minimally covers |
|
| Curves that minimally cover $X_{121p}$ |
|
| Curves that minimally cover $X_{121p}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 540t^{14} - 4536t^{12} - 20736t^{10} - 55755t^{8} -
88452t^{6} - 77436t^{4} - 30240t^{2} - 1728\]
\[B(t) = -54t^{24} - 1620t^{22} - 21708t^{20} - 171288t^{18} - 882981t^{16} -
3116718t^{14} - 7668486t^{12} - 13107420t^{10} - 15171624t^{8} - 11218608t^{6} -
4670784t^{4} - 767232t^{2} + 27648\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 3450x - 77875$, with conductor $1800$ |
| Generic density of odd order reductions |
$691/5376$ |